Ultrafast optical switching of three-dimensional Si inverse opal photonic band gap crystals
Tijmen G. Euser, Hong Wei, Jeroen Kalkman, Yoonho Jun, Albert Polman, David J. Norris, Willem L. Vos
aa r X i v : . [ phy s i c s . op ti c s ] S e p Ultrafast optical switching of three-dimensional Si inverse opalphotonic band gap crystals
Tijmen G. Euser
FOM Institute for Atomic and Molecular Physics,Kruislaan 407, 1098 SJ Amsterdam, The Netherlands
Hong Wei
Department of Chemical Engineering & Materials Science,University of Minnesota, Minneapolis, Minnesota 55455, USA
Jeroen Kalkman
FOM Institute for Atomic and Molecular Physics,Kruislaan 407, 1098 SJ Amsterdam, The Netherlands
Yoonho Jun
Department of Chemical Engineering & Materials Science,University of Minnesota, Minneapolis, Minnesota 55455, USA
Albert Polman
FOM Institute for Atomic and Molecular Physics,Kruislaan 407, 1098 SJ Amsterdam, The Netherlands
David J. Norris
Department of Chemical Engineering & Materials Science,University of Minnesota, Minneapolis, Minnesota 55455, USA
Willem L. Vos
FOM Institute for Atomic and Molecular Physics,Kruislaan 407, 1098 SJ Amsterdam, The Netherlands andComplex Photonic Systems (COPS), MESA + Research Institute,University of Twente, The Netherlands ∗ bstract We present ultrafast optical switching experiments on 3D photonic band gap crystals. Switchingthe Si inverse opal is achieved by optically exciting free carriers by a two-photon process. We probereflectivity in the frequency range of second order Bragg diffraction where the photonic band gapis predicted. We find good experimental switching conditions for free-carrier plasma frequenciesbetween 0.3 and 0.7 times the optical frequency ω : we thus observe a large frequency shift of upto ∆ ω/ω = 1.5% of all spectral features including the peak that corresponds to the photonic bandgap. We deduce a corresponding large refractive index change of ∆ n ′ Si /n ′ Si = 2.0%, where n ′ Si isthe refractive index of the silicon backbone of the crystal. The induced absorption length thatis longer than the sample thickness. We observe a fast decay time of 21 ps, which implies thatswitching could potentially be repeated at GHz rates. Such a high switching rate is relevant tofuture switching and modulation applications. PACS numbers: 42.70.Qs, 42.65.Pc, 42.79.-e . INTRODUCTION Currently, many efforts are devoted to an intricate class of three-dimensional metama-terials known as photonic crystals. Spatially periodic variations of the refractive indexcommensurate with optical wavelengths cause the photon dispersion relations to organize inbands, analogous to electron bands in solids. Generally, frequency windows known as stopgaps appear in which modes are forbidden for specific wave vectors. Experimentally, stopgaps appear as peaks in reflectivity spectra. The strong dispersion in photonic crystals canbe used to control the propagation direction of light. Fundamental interest in 3D photoniccrystals is spurred by the possibility of a photonic band gap, a frequency range for whichno modes exist at all. Tailoring of the photonic density of states by a photonic crystal al-lows one to control fundamental atom-radiation interactions in solid-state environments.
In this context the hallmark of a photonic band gap is the eagerly awaited inhibition ofspontaneous emission due to a vanishing density of states. Exciting prospects arise when 3D photonic band gap crystals are switched on ultrafasttimescales. First of all, switching the directional properties of photonic crystals leads to fastchanges in the reflectivity. Ultrafast control of the propagation of light, which was demon-strated in 2D photonic crystals by Ref. , is essential to applications in active photonic inte-grated circuits. Secondly, switching would also allow the capture or release of photons fromphotonic band gap cavities, which is relevant to solid-state slow-light schemes. Thirdly,switching 3D photonic band gap crystals provides dynamic control over the density of statesthat would allow the switching-on or -off of light sources in the band gap. An optical switching experiment requires a switching magnitude as large as possible, ul-trafast time-scales, as low as possible induced absorption, as well as good spatial homogeneityof the index change. In our experiment we change the refractive index of the semiconductorbackbone of the crystal by optically exciting a free-carrier plasma. The refractive index ofthe excited crystal is well described by the Drude model, in which the plasma frequency ω p is proportional to the density of excited carriers. By carefully choosing the amount of excitedcarriers, and thus the plasma frequency, large changes in refractive index are feasible, whilethe induced absorption is predicted to remain small. For Si, good experimental conditionsare found for free-carrier plasma frequencies around ω p = 0.3-0.7 × ω probe , where ω probe is thefrequency of the probe light. The spatial homogeneity of the excited carrier plasma can be3ptimized by choosing a two-photon excitation mechanism rather than a linear process. InSi photonic crystals, optimum spatial homogeneity is obtained for pump frequencies nearthe two-photon absorption edge of Si ω /c= 5000 cm − ( λ = 2000 nm). A pioneering study of switching 3D photonic materials was done by Mazurenko et al. , who reported reflectivity changes in silica opaline matrices that were infiltrated with Si. Thisexperiment suffered from several limitations; firstly, the limited refractive index contrast wasinsufficient for a band gap to open up. Secondly, since the experiments were performed atprobe frequencies above the electronic band gap of Si, the transparency of the unswitchedcrystal is limited by intrinsic absorption. Moreover, due to a short Drude damping time foramorphous Si τ Drude = 0.5 fs, the maximum feasible refractive index change is limited bythe amount of induced absorption. Original switching experiments in Si inverse opals werereported by Becker et al. who studied transmission changes. The induced absorption intheir crystal was strongly reduced by annealing the Si-backbone, causing τ Drude to increasedrastically from 0.5 fs to 10 fs, resulting in a strong reduction of the induced absorption.Unfortunately, however, this study was limited to the frequency range of first order Braggdiffraction where a pseudogap is expected, but not the photonic band gap.In this article we study ultrafast switching of inverse opal photonic band gap crystals.There are several reasons why inverse opals are highly suitable for all-optical switchingexperiments. Firstly, their fabrication is relatively simple, which has allowed inverse opalsto be studied extensively. The abundance of prior static reflectivity experiments helps usto interpret our switching data.
Secondly, the thickness of opaline crystals is notlimited by the fabrication process, in contrast to crystals that are grown by lithographictechniques. Thirdly, band structure calculations for inverse opals are easily available, facilitating the interpretation of the observed stop bands in our spectra. Fourthly, thecrystals can have a sufficiently large refractive index contrast for a band gap to open upin the range of second order Bragg diffraction, while in the range of first order Braggdiffraction a pseudo gap occurs. In the region of the band gap, switching is expectedto lead to ultrafast changes in the density of states. Finally, experimental control of thedynamics of spontaneous emission from quantum dots inside static photonic crystals wasrecently demonstrated with inverse opals. Thus, it has been predicted that the spontaneousemission of light sources inside such crystals can indeed be switched on and off. We thereforeexpect that a study of ultrafast switching inverse opals in the range of second order Bragg4iffraction is timely.
II. EXPERIMENTALA. Sample
The Si inverse opal photonic crystal was made by infiltrating Si in a silica opaline tem-plate. The template was grown on a Si wafer substrate by a vertical controlled dryingmethod. The resulting 3D silica template extends over 5 × ◦ C. Subsequently, the sample was annealed forone hour at 750 ◦ C in vacuum. During the annealing process, the amorphous Si in thestructure crystallizes into poly-Si, as was confirmed by Raman scattering measurements ona separately deposited thin layer of polycrystalline Si that served as a reference sample.
Finally, the SiO template was etched away by a buffered hydrofluoric solution, resulting ina high-quality 3D air-sphere crystal that is supported by a poly-Si backbone.We have obtained the microscopic structure of our on-chip Si inverse opal crystal fromhigh resolution scanning electron micrographs (SEM). From planar and cross-sectional SEMimages we infer that our inverse opal is a fcc crystal. The size of the lattice parameter of thefcc-lattice is obtained from Fig. 1, which shows a top view of a { } plane in the crystal.We find the lattice parameter to be a = 1427 ±
20 nm, by measuring the lattice parameteralong the three { h ¯ h } in-plane lattice directions, which allows us to correct for a 22 ◦ tiltof the sample. From the lattice parameter and the number of terrace steps, we deduce asample thickness of 7 × d = 7 × a / √
3= 5.8 µ m.We also observe from Fig. 1 that the interstices (indicated by arrows) in the { } surfaceappear to be small. To describe the crystal with band structure calculations, we use a modelof close packed air spheres (radius r in = a/ √
8) surrounded by spherical shells (radius r out )connected by cylindrical windows (radius r cyl ) see Ref. 14. From the almost filled intersticesin Fig. 1, we estimate an outer shell radius of r out = 1.15 ± r in . The radius of the airholes that interconnect the air spheres in the crystal are measured to be r cyl = 0.26 ± r in . This method results in a volume fraction of the solid material of about Φ Si = 21.3 %.Nevertheless, it should be realized that estimating volume fractions from SEM images canbe problematic, as was found in in-situ x-ray experiments. ǫ Si = 12.74 (at ω = 6535 cm − ) that was measured on a separately deposited ref-erence sample. The resulting photonic band structure calculation is plotted in Fig. 2(A).The open symbols in Fig. 2(B) represent a linear reflectivity spectrum of the sample. It isremarkable that the measured peaks correspond to the predicted stop gaps, since our modeldoes not include freely adjustable parameters.
B. Ultrafast switching setup
Our setup consists of a regeneratively amplified Ti:Sapph laser (Spectra Physics Hur-ricane) which drives two optical parametric amplifiers (OPA, Topas). Both OPAs have acontinuously tunable output frequency between 3850 and 21050 cm − , with pulse durationsof 150 fs and a pulse energy E pulse of at least 20 µ J. The independent tunability of ourOPAs allows us to optimize the pump frequency, while scanning the probe frequency overa broad frequency range. The pump beam is incident at θ = 15 ◦ , and has a much largerGaussian focus of 133 µ m full width at half maximum (FWHM) than the probe, providinggood lateral spatial homogeneity. The probe beam is incident at normal incidence θ = 0 ◦ ,and is focused to a Gaussian spot of 28 µ m FWHM at a small angular divergence NA=0.02. Therefore, we ensure that only the central flat part of the pump focus is probed. Thereflectivity was calibrated by referencing to a gold mirror. A versatile measurement schemewas developed to subtract the pump background from the probe signal, and to compensatefor possible pulse-to-pulse variations in the output of our laser, see Ref. 17. III. RESULTS AND DISCUSSIONA. Linear reflectivity spectra
The linear reflectivity spectra in Fig. 2(B) demonstrates three stops band in the frequencyrange ω = 5000-7000 cm − , similar to earlier work. We compare them to a calculatedband structure diagram, and label the stop bands. Stop band I at frequency ω = 5320 cm − displays a maximum reflectivity R= 51% and is identified with the Γ-L stop gap at frequencya/ λ = 0.76 in the calculated bandstructure shown in Fig. 2(A). At ω = 5950 cm − , we observe6 stop band, labeled II, with maximum reflectivity R= 60%. Stopband II corresponds to thestop gap at a/ λ = 0.85 in Fig. 2(A). It is important to note that stop band II overlaps thefrequency range of the predicted band gap that is centred around a/ λ = 0.85. Stop band IIIat ω = 6500 cm − aligns with the stop gap at a/ λ = 0.94. The frequency of the small peakat ω = 7616 cm − matches that of the stop band at a/ λ = 1.08.We have systematically reproduced our data on various positions on the sample. The peakreflectivity of all peaks varies by less than 10% with position. The variation is possibly dueto variations in the density of lattice defects throughout the crystal. The center frequencyof the stop bands, however, were found to be independent of position on the sample (within∆ ω/ω < B. Switched reflectivity spectra
We have induced large and ultrafast reflectivity changes in our crystal by optically ex-citing free-carriers. The ultrafast response of the stop bands is acquired by measuring thereflectivity spectra at fixed probe delays. In experiments on the red part of the spectrum( ω probe < − ), the pump frequency was chosen to be ω pump = 6450 cm − . In experi-ments on the blue edge (6250 cm − < ω probe ), the pump frequency was reduced to ω pump =5000 cm − . The switched reflectivity spectra were measured in the same run and on thesame spot on the sample as the linear data are shown as closed circles in Fig. 2(B). Due todispersion in the probe path of the setup, there is a <
500 fs variation of the delay timewith frequency, we therefore measure the switched reflectivity at a fixed time delay of τ ≈ ω = 5110 cm − , the re-flectivity strongly decreases, while on the blue edge, at ω = 5500 cm − we observe a strongincrease in the reflectivity, indicative of a blue shift of the entire peak. Our crystals aretherefore highly suitable to control the directional propagation of light.The magnitude of the frequency shift of the peaks was obtained by measuring the fre-quency position of the red edge of stop band I, indicated by the black arrow in Fig. 2(B).The switching moves the edge towards higher frequencies, while the slope of the stop bandedge remains unchanged. Only at the highest intensity used, induced absorption can slightlychange the slope of the edge. However, the contribution of this effect to the measured shiftis negligibly small.The blue-shift of peak I is as large as 80 cm − or ∆ ω/ω = 1.5%. The same effect occurson the red edge of peak II, near ω = 5800 cm − , where the reflectivity decreases, and on theblue edge of peak III, near ω = 7000 cm − , where the reflectivity increases. The blue edgeof stop band III has blue-shifted by 50 cm − or ∆ ω/ω = 0.7%. Importantly, all stop bandshave shifted towards higher frequency. We therefore conclude that switching has reducedthe average refractive index of the crystal.The shift of the stop bands is clearly evidenced by the dispersive features in the differentialreflectivity of the sample that is plotted in Fig. 2(C). On the red edge of stop band I, at ω = 5110 cm − , we observe a large decrease in the reflectivity by ∆R/R= -54%, while at ω =5500 cm − we observe a strong increase in the reflectivity by ∆R/R= 49%. This distinctdispersive shape that is centered at around 5320 cm − is related to a large blue shift ofstop band I; the observation of positive differential reflectivity indicates that the inducedabsorption remains small. Peak II and III are slightly broadened by disorder in the sampleand thus appear as a single peak in the spectrum. On the red edge of the combined peak,at ω = 5800 cm − the differential reflectivity is ∆R/R= -35%, while at the blue edge ofthe peak, at ω = 7020 cm − the differential reflectivity amounts to ∆R/R= +30%; herethe dispersive shape also has a strong positive component, which again signals low inducedabsorption. The strong dispersive shape that is centered around 6450 cm − is related to8 large blue shift of the combined stop bands II and III. The strong dispersive curve thatis centered around ω = 7600 cm − shows that the small peak at this frequency also shiftstowards higher frequency.The observed shift of stop band II towards higher frequency is particularly interesting, asthis stop band is part of the predicted band gap for inverse opals. We have thus demonstratedswitching of a 3D photonic band gap, which has not been reported before. The switchingprocess is expected to lead to ultrafast changes of the density of states inside the crystal. Remarkably, we observe that both the low and high frequency edge of the stop bands haveshifted. This indicates the absence of separate dielectric and air bands in the range of secondorder Bragg diffraction in inverse opals, which is consistent with predictions based on quasi-static band structure calculations by Ref. 6. From our comparison we find a refractive indexchange of ∆n’/n’ ≃ N eh = 2.1 × cm − . The correspondingplasma frequency is ω p = 3623 cm − , which is equal to 0.72 × ω probe . We conclude thatexcellent switching conditions indeed appear if the plasma frequency ω p remains smallerthan the probe frequency as predicted in Section I. C. Switching time traces
The large and ultrafast shift of the stop band with time is studied in detail in Fig. 3. Wehave measured the frequency position of the red edge of the stop band at ω = 5045 cm − , ata large range of delay times after excitation. From each spectrum, the frequency position ofthe low frequency edge was determined at R= 15% (indicated by the arrow in Fig. 2(B)).The relative frequency shift ∆ ω/ω is plotted versus probe delay in Fig. 3. We observe alarge and ultrafast shift of the stop band edge from 0 to ∆ ω/ω = 1.1% with an exponentialgrowth time of τ = 500 fs, limited by the pulse duration of our pump pulses.The effect subsequently decreases exponentially with a decay time of τ = 21 ± ω/ω = 0.1%. The decay times are much faster thancarrier relaxation times in bulk Si, since our photonic crystals are made of poly crystallinesilicon, whose lattice defects and grain boundaries act as efficient carrier recombinationtraps. The short relaxation time is in good agreement with the typical carrier relaxationtime of 18 ps that we found in poly-Si woodpile crystals. The relatively fast decay timeimplies that switching could potentially be repeated at GHz rates, which is relevant to9ossible future switching and modulation applications.
D. Induced probe absorption
Besides a frequency shift which is related to a change in the real part of the refractiveindex, we observe a decrease of the reflectivity peak, which is associated with an increasein n ′′ Si . The induced absorption manifests itself in a reduction of the reflectivity maximumafter excitation. We therefore plot in Fig. 4 the relative decrease in reflectivity maximumof stop band II at frequency ω = 5882 cm − as a function of delay time. The data wereobtained in the same run as the experiment shown in Fig. 3. The maximum decrease inreflectivity of the stop band directly after excitation, is ∆R/R ≈ -21%. Note that thisdecrease is not only due to absorption but also to a shift. In Ref. 17 we have related areduction in peak reflectivity of woodpile samples to the induced absorption through exactmodel method calculations. Due to difficulties in calculating the reflectance using a complexdielectric function in inverse opals, we estimate the induced absorption by comparing theobserved ∆R/R= -21% to the calculated decrease in Si woodpile crystals. For woodpilephotonic crystals theory that can handle a complex dielectric function is available. Fromthis comparison we estimate an upper bound to the induced absorption in the Si backboneto be n ′′ Si < .
1. We use the Si volume fraction Φ Si = 21.3%, obtained in II A, to describe ourcrystal as an effective medium consisting of Si and air. In more advanced studies, one couldalso take the spatial distribution of the probe light in the crystal into account. Since onlya fraction Φ Si of our crystal absorbs light, we can estimate the resulting probe absorptionlength in our switched inverse opal to be ℓ abs > λ/ π Φ Si n ′′ Si = 6.3 µ m. The obtained valueis larger than the thickness of the sample L= 7 × d { } = 5.8 µ m. We conclude that forrefractive index changes larger than 2%, n ′′ Si will increase further, and consequently thecrystal may lose its transparency. Likewise, in applications in which much smaller changesin the refractive index suffice, the induced absorption will become negligible small.Figure 4 also shows how the reflectivity change evolves in time after the initial decreaseto ∆ R/R = − R/R = −
10% with a decay time of4 ± cm − ), carrier recombination is dominated by Auger effects with recombination times ofthe order of 10 ns. Any change in the substrate is likely to change the magnitude of thereflectivity of the whole sample, while it should not affect the frequency positions of the stopbands in Fig. 2, which are related to changes in the backbone of the photonic crystal only.Indeed, we find that the stop band shift in Fig. 3 decays rapidly within 100 ps, to a smalloffset of a few wavenumbers. Meanwhile, a large part of the reflectivity decrease in Fig. 4 isstill present after 100 ps, and continues to decay on ns timescales, consistent with the slowrecombination of carriers in the wafer substrate.In our experiments we have used two-photon absorption at long pump wavelengths com-bined with a large pump focus to maximize the spatial homogeneity of the switched crystals.The pump fluence in our experiments was typically 25 pJ per unit cell area per pulse. Inapplications with small active areas, typical for 2D and 3D cavities, spatial homogeneity isnot an important issue and thus low pump fluences suffice. Furthermore, the pump fluencecan be further reduced by choosing the pump wavelength in the linear absorption range. IV. CONCLUSIONS
In this Paper, we have studied all-optical ultrafast switching of a high-quality 3D Si inverseopal photonic band gap crystal in the frequency range of second order Bragg diffraction. Aspatially homogeneous free-carrier plasma was optically excited in the crystal by a two-photon process. We show that for Si inverse opals, good experimental conditions are foundfor free-carrier plasma frequencies around ω p = 0.3-0.7 × ω probe ; large changes in the refractiveindex can be achieved, while the crystal remains transparent after the switching. We findgood agreement between the stop bands in the linear reflectivity spectra and calculated stopgaps in the frequency range of the band gap. Switching effects are studied as a functionof time delay between pump and probe pulses. Large ultrafast variations in reflectivity areobserved in the range of second order Bragg diffraction. During the switching process, allspectral features in the observed stop bands, shift towards higher frequencies by as much11s ∆ ω/ω = 1.5% within a few hundred fs, indicating the absence of separate dielectric andair bands in our crystal. From a comparison to quasi-static band structure calculations ofRef. 6 we infer a large refractive index change of about 2%. The deduced refractive indexchange is predicted to strongly modify the density of states inside the crystal. We haveobserved a relatively fast decay time of 21 ps, which implies that switching could potentiallybe repeated at a GHz rates, which is relevant to possible future switching and modulationapplications.
ACKNOWLEDGMENTS
We thank Cock Harteveld and Rob Kemper for technical support, Martin Wegener, AdLagendijk, Dimitry Mazurenko, and Patrick Johnson for discussions. This work was alsoreported on arXiv.org/abs/0705.4250. This work is part of the research program of the”Stichting voor Fundamenteel Onderzoek der Materie” (FOM), which is supported by the”Nederlandse Organisatie voor Wetenschappelijk Onderzoek” (NWO). This work was alsosupported in part by the ACS Petroleum Research Fund, the US NSF (CTS-0332484), andthe US DOE (DE-FG02-06ER46348). We also utilized the Nano Fabrication Center and theCharacterization Facility at the University of Minnesota which receive partial support fromUS NSF through the NNIN program. DJN acknowledges support from the Alexander vonHumboldt Foundation. 12
Electronic address: [email protected]; URL: ’Photonic Crystals and Light Localization in the 21st Century’, Ed. C. M. Soukoulis (Kluwer,Dordrecht, 2001). E. Yablonovitch, Phys. Rev. Lett. , 2059 (1987). P. Lodahl, A. F. van Driel, I. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, andW. L. Vos, Nature , 654 (2004). S. W. Leonard, H. M. van Driel, J. Schilling, and R. B. Wehrspohn, Phys. Rev. B , 161102(2002). H. Nakamura, Y. Sugimoto, K. Kanamoto, N. Ikeda, Y. Tanaka, Y. Nakamura, S. Ohkouchi,Y. Watanabe, K. Inoue, H. Ishikawa, and K. Asakawa, Opt. Express , 6606 (2004). P. M. Johnson, A. F. Koenderink, and W. L. Vos, Phys. Rev. B , 081102(R) (2002). M. F. Yanik and S. Fan, Phys. Rev. Lett. , 083901 (2004). K. Sokolowski-Tinten and D. von der Linde, Phys. Rev. B , 2643 (2000). T. G. Euser and W. L. Vos, J. Appl. Phys. , 043102 (2005). D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov,A. B. Pevtsov, and A. V. Sel’Kin, Phys. Rev. Lett. , 213903 (2003). C. Becker, S. Linden, G. von Freymann, M. Wegener, N. T´etreault, E. Vekris, V. Kitaev, andG. A. Ozin, Appl. Phys. Lett. , 091111 (2005). J. E. G. J. Wijnhoven and W. L. Vos, Science , 802 (1998). A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. L´opez,F. Meseguer, H. M´ıguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel, Nature , 437 (2000). W. L. Vos and H. M. van Driel, Phys. Lett. A , 101 (2000). Y. A. Vlasov, X- Z. Bo, J. C. Sturm, and D. J. Norris, Nature , 289 (2001). E. Palacios-Lidon, A. Blanco, M. Ibisate, F. Meseguer, and C. L´opez, Appl. Phys. Lett. ,4925 (2002). T. G. Euser, A. J. Molenaar, J. G. Fleming, B. Gralak, A. Polman, and W. L. Vos,http://arxiv.org/abs/physics/0603045 (2006). K. Busch and S. John, Phys. Rev. E , 3896 (1998). A. F. Koenderink, Emission and transport of Light in Photonic Crystals, ISBN 90-9016903-2,University of Amsterdam (2003). H. S. S¨oz¨uer, J. W. Haus, and R. Inguva, Phys. Rev. B , 13962 (1992). K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. , 3152 (1990). P. Jiang, J. F. Bertone, K. S. Hwang, and V. L. Colvin, Chem. Mater. , 2132 (1999). J. Kalkman, Controlled spontaneous emission in erbium-doped microphotonic materials, ISBN90-363-0295-2, University of Utrecht, (2005). J. Kalkman, E. de Bres, A. Polman, Y. Jun, D. J. Norris, D. C. t Hart, J. P. Hoogenboom, andA. van Blaaderen, J. Appl. Phys. , 2297 (2004). J. E. G. J. Wijnhoven, L. Bechger, and W. L. Vos, Chem. Mater. , 4486 (2001). P. Y. Yu and M. Cardona, Fundamentals of Semiconductors, Springer-Verlag, Berlin, (1996). H. W. Tan, H. M. van Driel, S. L. Schweizer, R. B. Wehrspohn, and U. G¨ o sele, Phys. Rev. B , 205110 (2004). J. Dziewior and W. Schmid, Appl. Phys. Lett. , 346 (1977). IG. 1: High resolution SEM image of the Si inverse opal after HF etching. The scale bar is 2 µ m.The arrow indicates an almost filled interstice in the structure. From this image we estimate theradius of the windows that interconnect the air spheres to be r cyl = 0.16 ± r sphere . IG. 2: (A) Photonic band structures for fcc close packed air spheres (radius r= a/ √
8) surroundedby spherical Si shells (radius 1.15) connected by cylindrical windows (radius 0.264r). The volumefraction of solid material is about Φ Si = 21.3% ( ǫ Si = 12.74). The frequency scale corresponds tothe one in (B) (C) for lattice parameter a= 1427 nm. The four light gray areas indicate stop gapsthat occur in the Γ-L direction. The dark gray bar indicates the frequency range of the band gap.(B) Unswitched (open squares) and switched (closed circles) reflectivity spectra of the sample atnormal incidence measured with our OPAs. The observed peaks in the frequency range of secondorder Bragg diffraction are labeled I, II, and III. In experiments on the red part of the spectrum( ω probe < − ), the pump frequency was chosen to be ω pump = 6450 cm − . In experiments onthe blue edge (6250 cm − < ω probe ), the pump frequency was reduced to ω pump = 5000 cm − . Thepump irradiance for the switched spectrum was I = 11 ± − on the red part and I = 24 ± − on the blue part of the stop band. The switched spectra were measured at a pump-probetime delay of ≈ ω = 5320 cm − is illustrated by strong decreases andincreases of the reflectivity below and above this frequency respectively. Similar dispersive curvesare centered at frequencies of 5960 (II), 6450 (III), and 7600 cm − (vertical dotted lines). IG. 3: Blue shift of the low frequency edge of stop band I plotted versus probe delay (symbols).The pump frequency and irradiance were ω = 6450 cm − and 4 ± − respectively. Thelarge shift amounts to ∆ ω/ω = 1.1% with an exponential growth time of τ = 500 fs (left-handpanel). The subsequent exponential decay is well fitted with a single exponential decay ∆ ω/ω =A+ exp( − t/τ ) (red curve), where the decay time τ = 21 ± ω/ω is A=0.13% (right-hand panel). IG. 4: Differential reflectivity at ω = 5882 cm − versus probe delay. The pump frequency andirradiance were ω = 6450 cm − and 11 ± − respectively. The large decrease amounts to∆R/R= -21% within the first 500 fs, followed by a decay that is well fitted with a single exponential∆ ω/ω = A+ Bexp( − t/τ ) (curve), with amplitude B= -11%, decay time τ = 4.5 ±0.5 ps and offsetA= -10%. The offset appears to decay slowly at ns times and is attributed to the wafer substrate.